Moduli theory associated to Hochschild pairs

Abstract: We consider an $A$-linear stable infinity-category $\mathcal{C}$ and the pair $(\mathcal{HH}^\bullet(\mathcal{C}/A),\mathcal{HH}_\bullet(\mathcal{C}/A))$ of the Hochschild cohomology spectrum (Hochschild cochain complex) and the Hochschild homology spectrum (Hochschild chain complex). The purpose of this paper is to provide a moduli-theoretic interpretation of the algebraic structure on the Hochschild pair of $\mathcal{C}$. The algebraic structure on the Hochschild pair is encoded by means of a two-colored topological operad called Kontsevich-Soibelman operad. The notions of cyclic deformations and equivariant deformations (of the Hochschild chain complex) associated to deformations of $\mathcal{C}$ play a central role.

• The paper introduces a moduli-theoretic interpretation of Hochschild pairs, linking them to deformations of stable ∞-categories via operadic and dg Lie algebra structures.
• It constructs moduli functors for cyclic and S¹-equivariant deformations, providing a precise bridge between categorical deformations and homological invariants.
• The work exploits functorial Koszul duality and explicit commutative diagrams to clarify the interplay between categorical and homological deformation theories.

Moduli-theoretic Interpretation of Hochschild Pairs

Introduction and Context

This paper establishes a deep connection between the algebraic structures arising from Hochschild homology and cohomology of stable $\infty$-categories and their deformation-theoretic moduli. By systematically leveraging higher category theory, operadic structures (notably the Kontsevich-Soibelman (KS) operad), and formal moduli problems controlled by dg Lie algebras, the work situates the Hochschild pair $$(\mathrm{HH}^*(\mathcal{C}/A), \mathrm{HH}_*(\mathcal{C}/A))$$ as the principal algebraic object encoding deformations of $A$-linear stable $\infty$-categories. The critical innovation is the introduction of moduli functors encapsulating "cyclic" and "$S^1$-equivariant" deformations, which bridge categorical and homological algebra.

Summary of Main Results

Algebraic Structures and the KS Operad

The paper constructs and promotes the Hochschild pair $(\mathrm{HH}^*, \mathrm{HH}_*)$ to an algebra over the KS operad, capturing both the $E_2$-algebra structure on Hochschild cochains and the $S^1$-action (via the Connes operator) on Hochschild chains. There is a canonical $E_2$-module action of Hochschild cohomology on Hochschild homology, formalized operadically.

Moduli Functors and Deformation Theory

The central technical advance is the moduli-theoretic interpretation of deformations via functors:

• The classical moduli functor $\mathrm{Def}_\mathcal{C}$: deformations of $\mathcal{C}$ parametrized by Artinian square-zero extensions.
• $\mathrm{Def}_{S^1}(\mathrm{HH}_*)$: moduli of $S^1$-equivariant deformations of the Hochschild chain complex.
• $\mathrm{Def}_\mathrm{cyc}(\mathrm{HH}_*)$: moduli of "cyclic deformations," a notion interpolating between ordinary and $S^1$-equivariant deformations.

There are natural transformations

$$\mathrm{Def}_\mathcal{C} \xrightarrow{\mathcal{M}_\mathcal{C}} \mathrm{Def}_\mathrm{cyc}(\mathrm{HH}_*) \xrightarrow{\mathcal{N}_\mathcal{C}} \mathrm{Def}_{S^1}(\mathrm{HH}_*)$$

with precise categorical and derived interpretations.

Lie Theoretic Descriptions and Formal Stacks

Deformation functors are shown to be governed, in the formal sense, by dg Lie algebras:

• The deformation theory of $\mathcal{C}$ is controlled by the dg Lie algebra $\mathfrak{g}_\mathcal{C}$ arising from the $E_2$-algebra structure of $\mathrm{HH}^*(\mathcal{C}/A)$.
• $S^1$-equivariant deformations of $\mathrm{HH}_*$ are governed by the dg Lie algebra $\operatorname{End}(\mathrm{HH}_*)^{S^1}$.

Kan extensions and Koszul duality arguments provide fully faithful functors between the corresponding formal moduli stacks, revealing that the deformation theory of the category and of its Hochschild chains are tightly linked via explicit Lie algebra maps produced by the KS algebraic structure.

Commuting Diagrams and Koszul Duality

The main theorem (Theorem 1.2, Theorem 8.23) asserts the existence of a commutative diagram (up to homotopy) of functors between deformation groupoids, formal stacks, and certain endomorphism Lie algebras, which is entirely governed by the algebraic structure of the Hochschild pair. Furthermore, the sequence of functors relating deformations, cyclic deformations, and equivariant deformations is shown to factor through explicit duality constructions, generalizing classical Koszul duality and making essential use of the bar construction on $E_n$-algebras.

Key Technical Tools

• Operadic formalism: The KS operad underlies simultaneous $E_2$-algebra and $S^1$-module action structures, encoding all relevant algebraic data for the Hochschild pair.
• Formal moduli theory: In the sense of Lurie and Hennion, deformation functors (over square-zero extensions) are represented by formal stacks, which can themselves be described entirely in terms of dg Lie algebras via Koszul duality.
• Factorization through cyclic deformations: The newly introduced moduli functor of cyclic deformations provides the essential intermediate step for relating category-level deformations to those of the Hochschild chain complex, illuminating the precise role of the cyclic and $S^1$-equivariant structures.
• Functorial Koszul duality: The paper employs functorial Koszul duality at the level of monoidal $\infty$-categories and demonstrates its compatibility with Hochschild functors, yielding explicit comparisons and equivalences between different flavors of deformation groupoids.

Applications and Implications

Noncommutative Hodge Theory and Period Maps

One major application is the construction of period maps for families of stable $\infty$-categories in noncommutative algebraic geometry. The negative and periodic cyclic homology, together with their Hodge-type filtrations, are accessible via the functorial moduli description developed here. The explicit linkage between cyclic deformations and the classifying stack of filtrations shows promise for extending classical Hodge-theoretical invariants to the noncommutative setting.

Higher Deformation Theory and Derived Algebraic Geometry

The explicit description of the deformation theory of stable $\infty$-categories in terms of the algebraic structures on their Hochschild (co)homology opens avenues for higher deformation quantization and the study of derived Azumaya algebras and D-modules in the categorical setting. These tools are directly relevant for subjects such as the categorification of the period map and the structure of D-modules on the moduli space of objects in a stable category.

Theoretical Consequences

The identification of the key controlling objects—dg Lie algebras and their actions (especially $S^1$-equivariant ones)—provides conceptual clarification of long-standing relationships between categorical and homological deformation theories. It also strengthens the connection to derived algebraic geometry, solidifying the bridges to factorization homology, topological field theory, and the study of integral transforms in categorical representation theory.

Conclusion

This work provides a unified, operadically-informed framework for understanding deformations of stable $\infty$-categories and their Hochschild pairs, attaching to the latter a precise and computable moduli-theoretic interpretation. It integrates higher algebraic and homotopical structures, leveraging formal moduli theory and Koszul duality to produce a suite of canonical equivalences and commutative diagrams relating disparate deformation problems. The explicit functoriality and compatibility with homological invariants developed here position the results as foundational for further advances in noncommutative geometry, derived deformation theory, and higher category theory.